Lemma 19.13.4 (07D9)—The Stacks project

Lemma 19.13.4. Let $\mathcal{A}$ be a Grothendieck abelian category. Then

$D(\mathcal{A})$ has both direct sums and products,
direct sums are obtained by taking termwise direct sums of any complexes,
products are obtained by taking termwise products of K-injective complexes.

Proof. Let $K^\bullet _ i$ , $i \in I$ be a family of objects of $D(\mathcal{A})$ indexed by a set $I$ . We claim that the termwise direct sum $\bigoplus _{i \in I} K^\bullet _ i$ is a direct sum in $D(\mathcal{A})$ . Namely, let $I^\bullet$ be a K-injective complex. Then we have

$\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(\bigoplus \nolimits _{i \in I} K^\bullet _ i, I^\bullet ) & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(\bigoplus \nolimits _{i \in I} K^\bullet _ i, I^\bullet ) \\ & = \prod \nolimits _{i \in I} \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(K^\bullet _ i, I^\bullet ) \\ & = \prod \nolimits _{i \in I} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(K^\bullet _ i, I^\bullet ) \end{align*}$

as desired. This is sufficient since any complex can be represented by a K-injective complex by Theorem 19.12.6. To construct the product, choose a K-injective resolution $K_ i^\bullet \to I_ i^\bullet$ for each $i$ . Then we claim that $\prod _{i \in I} I_ i^\bullet$ is a product in $D(\mathcal{A})$ . This follows from Derived Categories, Lemma 13.31.5. $\square$

Comments (2)

Comment #9963 by Elías Guisado on February 04, 2025 at 06:06

One could add either as a fourth property of as a separate new result:

Lemma. Let $\mathcal{A}$ be a Grothendieck abelian category satisfying AB4*. Then products in $D(\mathcal{A})$ are obtained by taking termwise products of any complexes.

Proof. Continuing at the end of the proof of Lemma 19.13.4, we have that $\prod K_i^\bullet\to\prod I_i^\bullet$ is a quasi-isomorphism since $\mathcal{A}$ is AB4* and a K-injective resolution by Derived Categories, Lemma 13.31.5. We are done. $\square$

Comment #10112 by Elías Guisado on April 23, 2025 at 03:55

@#9963 Sorry, apparently this result was already included: it's 13.34.2.

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